Integration of a Total Differential

If the coefficients of the total differential of a dependent variable are known as functions of the independent variables, the expression for the total differential may be integrated to obtain an expression for the dependent variable as a function of the independent variables. 

For example, suppose the total differential of the state function f x, y,z) is given by Eq. F.2.2 and the coefficients are known functions a x,y,z), b x,y,z), and c x,y,z). Because / is a state function, its change between /(0,0,0) and f x, y, z ) is independent of the integration path taken between these two states. A convenient path would be one with the following three segments  

Here is an example of this procedure applied to the total differential 

In chemical thermod5mamics, there is not likely to be occasion to perform this kind of integration. The fact that it can he done, however, shows that if we stick to one set of independent variables, the expression for the total differential of an independent variable contains the same information as the independent variable itself. 

A different kind of integration can be used to express a dependent extensive property in terms of independent extensive properties. An extensive property of a thermodynamic system is one that is additive, and an intensive property is one that is not additive and has the same value everywhere in a homogeneous region (Sec. 2.1.1). Suppose we have a state function / that is an extensive property with the total differential 

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The integral of a total differential, however, is the difference between two values of a state function and, therefore, cannot depend on the path of the integral. This holds even if this difference is evaluated by means of partial derivatives. 

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